## Two Planes

I stumbled upon the following problem in *Mathematics Teacher* v.73 (September 1980):

A plane takes off and goes east at a rate of 350 mph. At the same time, a second plane takes off from the same place and goes west at a rate of 400 mph. When will they be 2000 miles apart?

Ooh, boy!

Question for my readers: explain my reaction.

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## Vortico:

I believe it’s 8/3 hours after the take-off, assuming the Earth is flat. What’s the catch?

7 January 2011, 5:56 pm## Sachin Shanbhag:

At what latitude?

7 January 2011, 6:48 pm## Leo:

@Sachin: Also, straight line distance or distance on a sphere?

7 January 2011, 7:01 pm## Xamuel:

Time zones?

7 January 2011, 7:04 pm## Khalil Sawant:

If they are on the same run-away, better call the Emergency Airport Fire Services

8 January 2011, 12:57 amThis is a May-Day !!!

## Brenda:

1. Do we also assume the planes accelerate to their cruising speed instantly, and fly just above the ground?

8 January 2011, 1:47 am2. As pointed out, what latitude? For short distances, the curvature of the Earth would be close enough to a plane to be ignored, but 2000 miles is a long way around the sphere.

3. And yeah, it’s an odd airport that has planes leave in opposite directions at the same time. They must have a really exciting traffic pattern.

4. How would the distance be measured? Dead reckoning? From the viewpoint of an unmoving observer?

## Philip Gibbs:

I think it would be reasonable to ignore the curvature of the Earth, altitude changes, and relativistic effects. The student should be able to see that these effects are too small to be worth taking into account. They should also see that they are expected to ignore the fact that real planes accelerate

8 January 2011, 3:06 amHowever, since the question asks “when” it should give the time of departure so that the answer can be a specific time. If they want the answer “after 2 hours 40 minutes” they should ask “after how long”. I think most people would figure out what was required, but giving a departure time would make it a cleaner question.

## Sachin:

@Leo, I think East-West is defined along a latitude. So as you go away from the equator towards the poles, the “circumference” along a latitude decreases to zero as one approaches the poles. So above a certain latitude, the planes would collide.

If the problem said, north-south, instead of east-west, the same problem wouldn’t exist.

8 January 2011, 9:24 am## S.L.:

mph is a rate, so the problem says rate of a rate which would be something like the acceleration rather than the speed?

8 January 2011, 9:53 am## Christ Schlacta:

I’m going to have to say “Not enough information”. we don’t know too many variables to offer even a reasonable equation to express the answer. we don’t know when the plans left, we don’t know their rate of acceleration, we don’t know their latitude or their altitude, for all we know they could be 50 miles from the pole in which case they’ll never be 2000 miles apart. finally, we don’t know how large the airport is or how far apart the two runways are that these plans departed from.

8 January 2011, 4:09 pm## Nathan:

Two planes take off from the same place at the same time, thus occupying the same space at the time of take off? I would like to see that.

8 January 2011, 7:59 pm## colorblind:

“When” vs. “After how long” seems to be the correct answer to the cause of your reaction. Of course, given the use of a present tense, “A plane takes off”, perhaps we can assume our relative now is the time the plane is taking off. It’s about 1:17 AM here, so MY answer would be 3:57 AM. Your answer may be different.

On another note, at no point do these hypothetical planes stop flying. After some point, the airplanes will go so far around the sphere that they become closer and closer until they cross paths again. Because of that, there actually may be an infinite number of answers to the problem dependent on a poorly defined earth.

9 January 2011, 2:19 am## Walking Randomly » Carnival of Mathematics #73 – Chuck Norris Edition:

[…] her post, Two Planes, Tanya Khovanova stumbled across a seemingly innocent looking question in an old edition of […]

9 January 2011, 8:48 am## Tanya Khovanova’s Math Blog » Blog Archive » Two Planes Keep Flying:

[…] days ago I threw at my readers the following problem: A plane takes off and goes east at a rate of 350 mph. At the same time, a second plane takes off […]

9 January 2011, 4:12 pm## erik:

Is 2 hours and 40 minutes be the correct answer? The distance between the planes would be 750 miles per hour.

12 July 2011, 10:03 am